There is no way you can build up an acceptable sporting rifle on any military receiver for less than you can buy a factory built sporter that is already chambered in the cartridge you mention.
Do a bit of looking around. The internet is your friend. So are gun shows. A Remington or Winchester in 338WM can be had for less than $600.
Even if you got that action as a freebie, it would cost $400 for a barrel blank, minimum of $150 for a Boyd's stock, $200 to get a gunsmith to chamber, fit and headspace the barrel properly. Then if you decide you need a different action, you might get lucky and find just what you're looking for with all of the trimming done for $100+ Then, there is the cost of a blue job, add another $150.
A new in the box Savage can be had in the $750 range, stainless and in a walnut stock. P&D Enterprises. IMHO Savage rifles are fugly but do they ever shoot well. It's all built, ready to go and on a very decent platform, for what it will cost you to get the milsurp up and running and it will be backed up by a factory warranty as well.